Math Course Advice -- Harvard freshman planning to double major in physics and mathematics

[Quantum55151]

Hi everyone!

I would like to get advice concerning my math course for the spring semester.

For context, I am a Harvard freshman planning to double major in physics and mathematics with the long-term goal of doing research in high-energy theory and/or mathematical physics. I am currently finishing math 25A, a proof-based linear algebra course. For the spring semester, I have the option of either taking 25B, a proof-based real analysis course, or moving up to Harvard's infamous math 55. In the spring, math 55B covers topology (both point-set and algebraic) and complex analysis with a brief two weeks of real analysis in between.

Now, 25B is an objectively easier class, because it attempts to cover much less material in one semester than does 55B and also assumes less mathematical maturity on the part of students. For me at least, this might translate into a better understanding of real analysis than what I would gain from 55B where the professor tries to speed run through Rudin over the course of...2 weeks lol. At the same time, based on what I've read (and please correct me if I am wrong), I don't think real analysis is particularly useful for theoretical physics (at least not the kind that is taught in 25B; integration and measure theory might be a different story, but that is taught in a different class altogether). Topology and complex analysis, on the other hand, seem to be much more relevant to the kind of physics that I want to do. The other nice thing about 55B is that it would allow me to save time by basically knocking out three undergrad math courses in one semester which in turn would allow me to take more advanced undergrad classes and/or grad classes sooner. On the flip side, the trade-off will consist in how well I will actually learn the material in 55B as well as the level of difficulty of the class which, although a far cry from the stuff you'll read on 55's Wikipedia page, is nevertheless non-negligible.

What do you think? At any rate, I can always try out 55B for the first few weeks and then drop down to 25B if necessary. But I would still appreciate any advice so that I have a better idea of my spring plans and can plan out my winter studies accordingly.

P.S. I am attaching the syllabi for the two courses in case anyone wants to take a look.

 

[jedishrfu]

In college in the 1970s, I took a topology course, thinking I knew how to do proofs since I had taken geometry in high school, so it wouldn't have been a big deal.

1970s Geometry was taught a bit differently than it is now. Everything we learned started with axioms, then lemmas and then theorems. we proved everything. I thought I was prepared.

I skipped past freshman Set Theory and Abstract Algebra and jumped into Algebraic Topology. I was a junior physics major amid a sea of senior math majors, and set theory and Abstract Algebra were not required for physics majors.

One of my professors introduced me to the notion that if you throw enough mud at the wall, some of it will stick. Sadly, my wall was made of Teflon because nothing stuck. Each point set topology proof required knowing the definitions of many math terms and prior theorems, and I got lost in an ocean of words. It was like being a lawyer who never went to law school.

---

The professor who taught topology had also been my Calculus I professor. After questioning me on the basics of the course, he allowed me to skip ahead to Calculus II. I sailed through his questions until I hit the definition of a limit.

I had a physics major mindset: Math was for solving physics problems, so why care about the foundational issues of limits if you could differentiate any function?

After a couple of sessions of trying to explain limits in my own words, I memorized the limit definition. Then he said, "I think you got it." Let me skip ahead.

---

The professor felt some compassion for my hubris and helped me survive the onslaught so that I emerged wounded but whole. I struggled epically through the course, got a C, and was traumatized by the experience.

The C lowered my GPA, which was slowly going downhill as the courses got tougher. I learned that it's not good to be in a hurry. Enjoy your studies and soak up everything, look deeper into the material, and follow the well-trodden path of other students.

---

Know your abilities, and don't take unnecessary risks.

 

[PeroK]

What do you think? At any rate, I can always try out 55B for the first few weeks and then drop down to 25B if necessary. But I would still appreciate any advice so that I have a better idea of my spring plans and can plan out my winter studies accordingly.

I think you've summed things up yourself quite well. 55B looks very ambitious to me. There must be a risk that you'll crash and burn with that material. Even 25B is not trivial, but looks more plausible.

It's difficult for me to see how 55B would make much sense without 25B. It looks like starting GR before you've learned SR. It's true that Real Analysis, per se, is not necessary for theoretical physics, but it would give you a huge insight into concrete, pure mathematical thinking and logic. It would allow you to make much more sense of infinite sequences, series, functions etc. - including where physics cuts the corners. Otherwise, you may see more advanced mathematical material as more of a box of tricks.

 

[CrysPhys]

What do you think? At any rate, I can always try out 55B for the first few weeks and then drop down to 25B if necessary. But I would still appreciate any advice so that I have a better idea of my spring plans and can plan out my winter studies accordingly.

Given that you are asking about specific courses at your specific university, have you talked to your advisor and the professors who will be teaching the courses?

 

[Quantum55151]

I think you've summed things up yourself quite well. 55B looks very ambitious to me. There must be a risk that you'll crash and burn with that material. Even 25B is not trivial, but looks more plausible.

Yes, that is a very real risk and one that I fear. 25B is definitely not trivial, but after excelling in 25A, I think 25B would be quite manageable.

It's difficult for me to see how 55B would make much sense without 25B. It looks like starting GR before you've learned SR.

Technically, it's an either-or. In theory, 55B is supposed to cover all of the material of 25B, so 55B is like a superset of 25B. In practice, a lot of the real analysis material is sacrificed in favor of topology and complex analysis.

It's true that Real Analysis, per se, is not necessary for theoretical physics, but it would give you a huge insight into concrete, pure mathematical thinking and logic. It would allow you to make much more sense of infinite sequences, series, functions etc. - including where physics cuts the corners. Otherwise, you may see more advanced mathematical material as more of a box of tricks.

That is definitely a valid argument. Thank you!

 

[Quantum55151]

Given that you are asking about specific courses at your specific university, have you talked to your advisor and the professors who will be teaching the courses?

My advisor doesn't know anything about STEM courses; she just signs off my schedule. The respective professors have given me somewhat contradictory advice: the 55B professor told me to definitely try out his course, while the 25B professor said that, while I could definitely give 55B a shot, he thinks that cramming so much material into so little time is not a good idea. So I don't know...

 

[Quantum55151]

Enjoy your studies and soak up everything, look deeper into the material, and follow the well-trodden path of other students.

---

Know your abilities, and don't take unnecessary risks.

Thank you for your advice!

 

[jedishrfu]

If you take the 25 course first, which is included in the 55 course, you will have the opportunity to learn 25's material more deeply.

This will allow you to manage your grades better when taking 55 because you will have lessened your course load, having already covered a portion of what's in 55.

And you're getting credit for two courses. A win-win situation!

"Be like water my friend." -- Bruce Lee

 

[Quantum55151]

If you take the 25 course first, which is included in the 55 course, you will have the opportunity to learn 25's material more deeply.

This will allow you to manage your grades better when taking 55 because you will have lessened your course load, having already covered a portion of what's in 55.

Unfortunately, one can only take 55 as a freshman, so if I take 25 in the spring, then I won't be able to take 55 the following semester. So it's a choice between 55 and 25 + the "normal" undergrad courses that cover topology and complex analysis.

 

[jedishrfu]

Wow, that makes it more challenging to decide.

My son had this problem in high school. They took a two-year course that combined biology, chemistry, and physics. It was an IB-level course, and you could get real college credit for taking and passing it. The problem was the accelerated nature of the course. If you dropped out at any point, you had nothing to show for your effort.

He had to decide this during his freshman year of high school. He started, realized it was too fast-paced, and dropped out early.

If you want to risk it, then you could take it and drop it early without too much grief.

 

[Frabjous]

What does the prof who taught you 25A say?

 

[Muu9]

i can only imagine 55b assumes (in practice if not in theory) the level of background provided by 55A, rather than 25A. Maybe I'm wrong, in which case shopping 55b would help.

You could also ask the Harvard subreddit.

Another question: What physics classes did you take first semester and what do you plan to take next semester?

https://www.math.harvard.edu/media/Courses-in-Mathematics.pdf - could you forego 25b for some combination of 113, 122, and 131? Those seem like they would be more relevant, and I can't imagine taking 2 in one semester would be less rigorous than 25B.

IMO, it would be a choice between 55B and some combination of 113, 112, and 131, if that's possible

 

[mathwonk]

@Quantum55151: I am a (retired) pure mathematician, and that is my perspective. So my orientation is to want to learn the mathematics as deeply as possible, without interest in physics.
With that given, when I was in college I made my choices based on a factor you seem to ignore, namely who are the professors and what is their background and/or expertise, as well as their reputations as teachers. In this case, the 55B prof, Denis Auroux, is a world renowned pure mathematician holding a named Harvard professorship, i.e. a superstar mathematician. This is the kind of pedigree associated with teaching this course, which attracts the strongest most ambitious students.
The 25B prof, John Cain, is no doubt a very strong (applied) mathematician, but he is a lecturer, not at all the same level of faculty at Harvard.

I think you will be well served by both, but this is a chance to see whether you notice the difference, and to receive instruction from the sort of scholar you will possibly never see again in a classroom. Especially since Prof Auroux himself urges you to at least try his class, I suggest going for a look, and finding out if it is life changing in terms of quality. In that case, make sure to respect how much time it takes to complete a course that contains all of 25B and more, and adjust your other courses accordingly. If 55 is just overwhelming, you can drop back, and even possibly still audit it.

When I took math 55 from Loomis, I simultaneously audited 112 from Mackey to help fill in the metric space background 55 did not then contain. I never got credit for 112 but I did learn the material. (Loomis was also a very careful lecturer who wrote everything clearly on the board, and tended to cover less than other math55 professors.). I was also a junior (probably the oldest student in the class) and had already taken linear algebra, a big help.

Don't overwhelm yourself and drown, but you might regret missing out on a chance to hear what Prof Auroux's course sounds like.

If you take 55, be prepared for the most challenging course you have ever had, mainly because it goes ridiculously fast, and consider forming a study partnership with at least one other person, to help each other. Don't make the mistake of thinking, because you are a good student, and have always found math courses fairly easy or manageable alone, that you won't benefit from help.

If you have time now, you might audit a bit of 55A in preparation, but it may be essentially over. I recall one of my more savvy classmates used to audit the hard courses he aspired to before taking them. (You could also look at a copy of the math55a notes over vacation. I actually do not see much in the algebra that is covered in 55a that will be required for 55b, but Prof Auroux is the one to ask what exactly you should do to prepare.)

And please don't be bullied by me or anyone else, into signing up for a course that does not mean enough to you personally, to pay the high price of admission. I also think math 55, in these days, makes a serious pedagogical mistake, by cramming way too much material into much too short an amount of time to allow mastery.

good luck! you are in an enviable position, two good choices beckon.

And if you take 55, I suggest you get ahold of and consult, not only copies of the old course notes online, but also some books that are well written and explain the analysis and topology clearly, like books by Berberian, Cartan, Wallace, Munkres, Massey,.... And before finals, practice on copies of old exams, (formerly) available in house libraries.

 

[mathwonk]

And since you have me thinking about it, here is a synopsis of analysis, i.e. function theory. There are basically 4 kinds of (progressively more restrictive) functions, hence 4 kinds of analysis, 1) measurable/integrable functions, 2) continuous functions, 3) smooth functions, 4) analytic functions.

1) Real analysis/integration theory is about the first kind, where individual values don't matter much, just so the function can be averaged nicely over a region. I know little about this subject, but very choppy and bad looking functions can have nice averaging properties.
2) Topology is about continuous functions f, functions so that at each point p, the value f(p) is completely determined by the values f(x) for x near p. These have nice properties, like the intermediate value theorem. They preserve key ideas like connectedness and compactness; in particular they carry loops to loops.
3) Real calculus is about smooth functions, i.e. those having infinitely many continuous derivatives, which only makes sense on nice spaces:"manifolds", that look locally like Euclidean space. These are locally approximated by linear functions (that's the main fact I learned in math55a), and the main theorems are inverse and implicit function theorems, and Green/Stokes theorems (relation of area integrals to path integrals).
4) Complex analysis is about analytic functions, i.e. those locally represented by power series. Functions in complex calculus, i.e. functions locally approximated by complex linear functions, are actually also represented by power series. The famous Cauchy theorem is just Green's theorem carried over, but the analyticity result gives a refinement allowing computation of path integrals by "residues", i.e. certain coefficients of the power series.
One also gets the "identity" theorem, that a complex analytic function's values are determined everywhere in a connected region just by the values on one convergent sequence!
Complex analysis and Riemann surfaces benefit greatly from knowledge of topics from topology like loop ("fundamental") groups, and covering spaces (since complex differentiable maps from C≈R^2 to itself, look locally like the complex power functions z-->z^n).

 

[WWGD]

I am a (retired) pure mathematician, and that is my perspective. So my orientation is to want to learn the mathematics as deeply as possible, without interest in physics.
With that given, when I was in college I made my choices based on a factor you seem to ignore, namely who are the professors and what is their background and/or expertise, as well as their reputations as teachers. In this case, the 55B prof, Denis Auroux, is a world renowned pure mathematician holding a named Harvard professorship, i.e. a superstar mathematician. This is the kind of pedigree associated with teaching this course, which attracts the strongest most ambitious students.
The 25B prof, John Cain, is no doubt a very strong (applied) mathematician, but he is a lecturer, not at all the same level of faculty at Harvard.

I think you will be well served by both, but this is a chance to see whether you notice the difference, and to receive instruction from the sort of scholar you will possibly never see again in a classroom. Especially since Prof Auroux himself urges you to at least try his class, I suggest going for a look, and finding out if it is life changing in terms of quality. In that case, make sure to respect how much time it takes to complete a course that contains all of 25B and more, and adjust your other courses accordingly. If 55 is just overwhelming, you can drop back, and even possibly still audit it.

When I took math 55 from Loomis, I simultaneously audited 112 from Mackey to help fill in the metric space background 55 did not then contain. I never got credit for 112 but I did learn the material. (Loomis was also a very careful lecturer who wrote everything clearly on the board, and tended to cover less than other math55 professors.). I was also a junior (probably the oldest student in the class) and had already taken linear algebra, a big help.

Don't overwhelm yourself and drown, but you might regret missing out on a chance to hear what Prof Auroux's course sounds like.

If you take 55, be prepared for the most challenging course you have ever had, mainly because it goes ridiculously fast, and consider forming a study partnership with at least one other person, to help each other. Don't make the mistake of thinking, because you are a good student, and have always found math courses fairly easy or manageable alone, that you won't benefit from help.

If you have time now, you might audit a bit of 55A in preparation, but it may be essentially over. I recall one of my more savvy classmates used to audit the hard courses he aspired to before taking them. (You could also look at a copy of the math55a notes over vacation. I actually do not see much in the algebra that is covered in 55a that will be required for 55b, but Prof Auroux is the one to ask what exactly you should do to prepare.)

And please don't be bullied by me or anyone else, into signing up for a course that does not mean enough to you personally, to pay the high price of admission. I also think math 55, in these days, makes a serious pedagogical mistake, by cramming way too much material into much too short an amount of time to allow mastery.

good luck! you are in an enviable position, two good choices beckon.

And if you take 55, I suggest you get ahold of and consult, not only copies of the old course notes online, but also some books that are well written and explain the analysis and topology clearly, like books by Berberian, Cartan, Wallace, Munkres, Massey,.... And before finals, practice on copies of old exams, (formerly) available in house libraries.

I think there are YouTube videos of D Auroux, thennat MIT, teaching Advanced Calculus.

 

[jedishrfu]

That was a fantastic analysis @mathwonk. I wish you had been around when I was shepherded into taking Algebraic Topology at Union College circa 1973. It got me thinking about my prof and how overwhelmed I was by the course definitions, theorems, and proofs.

A key part of my confusion was that I couldn't relate to the material, having only a vague idea that it could be applied to black holes but never really seeing how. This can happen in fast-paced courses where you can't keep up. Especially in the pre-internet days, when you had to find the right books and trudge around finding and talking to the right people; that was just too much work for a student to do while working 20 to 30 hours per week (I know, crazy, but I had no debt when I got out).

---

My professor at the time was Professor William Fairchild. I was trying to remember the book, but upon researching Prof Fairchild's pedigree, I discovered that he and another Union professor had written a book on Topology that we used in class.

The book's title was, you guessed it, "Topology." It was written by Fairchild and Tulcea, both of Union at the time, and published by Saunders. The book is out of print but available on Abebooks.com.

---

My one question to you is: Have you heard of Prof. Fairchild, Tulcea, or their book?

Prof. Fairchild graduated from the University of Illinois at Urbana-Champaign in 1967. I recall that he got tenure rather quickly because there was a serious commotion over a student's favorite professor who taught there for a few more years but didn't get tenure.

It had to do with some campus politics where the favored professor had opposed using another professor's book in the set theory course he taught (lost money/bruised ego -- I guess). There was also a change of departmental focus from the WW2 war years of teaching math to engineers to doing math research, published with teaching as a side hustle.

The favored professor devoted his efforts to teaching rather than publishing papers, so his contract was not renewed. Later, he became a respected math department chair at another college. His Rate My Professors rating was also always very high.

---

The old math chairman, a great and respected teacher (Prof William Stone), stepped down, and a more progressive Prof Seiken took the reins. The change gave the department the chance to hire higher-quality math talent with research chops, which I think Prof Fairchild, a new hire and book author, had.

---

I miss those days and wish I could have devoted my full time to academics, but I was anxious and in a hurry, concerned about college debt, the Vietnam War, and other worldly turmoil. Also, I was on the fence between physics and math and couldn't decide, although now I realize I could have become a geometer or applied mathematician.

 

[Quantum55151]

What does the prof who taught you 25A say?

He says that I could definitely give 55B a try, but in his view, 55B is too superficial in its treatment of the material, especially of real analysis.

 

[jedishrfu]

I found this article on Prof Auroux and Math 55 at Harvard:

https://www.thecrimson.com/article/2023/3/26/behind-math-55/#:~:text=Denis Auroux, a professor who,is completely fine for you.”

What We Talk About When We Talk About Math 55​

Just five years ago, the Math Department’s official word on Math 55 was that it was “probably the most difficult undergraduate math class in the country.” Now, they say, “if you’re reasonably good at math, you love it, and you have lots of time to devote to it, then Math 55 is completely fine for you.” So, what changed?

BY SAGE S. LATTMANMarch 26, 2023
On a Wednesday morning on the fifth floor of the Science Center, 20 or so students mill about while Professor Joseph D. Harris ’72 draws on the chalkboard. Harris has sketched a “three-sheeted covering space,” a looping squiggle.

Harris spends the next hour and 15 minutes of Math 55B: “Studies in Real and Complex analysis” lecturing on the classification of covering spaces. He draws arrows and circles and greek letters on the board. He discusses topologies and neighborhoods and endpoints.

Thirty minutes in, Harris pauses drawing his diagrams, which by now take up the entire chalkboard, and turns to face the class. “Now we get to have some fun,” he says.

Welcome to Math 55, the undergraduate course surrounded by what is perhaps the most intrigue and infamy of any class at Harvard.

 

[Quantum55151]

The 25B prof, John Cain, is no doubt a very strong (applied) mathematician, but he is a lecturer, not at all the same level of faculty at Harvard.

I've actually thought about this. In fact, I took 55A for a week and then dropped (I had zero proof-writing or abstract mathematics experience and had never taking a proof-based course before, so I was just unprepared for the rigors of 55; there is hope that 55B might work out for me after a semester of 25A). However, that one week of 55A was enough for me to see how much of a "superstar mathematician" Auroux is, to use your words. This is something that I missed throughout the rest of my semester in 25A. So yes, the two professors will definitely be a factor in my choice.

If you take 55, be prepared for the most challenging course you have ever had, mainly because it goes ridiculously fast

Right, so the question is, do I really need this most-challenging-course-I've-ever-had experience and one that, as you say, is not the most pedagogically effective? Because after all, I am first and foremost a physics major, math being my secondary field. So, while definitely helpful for theoretical/mathematical physics, I am not sure how crucial/indispensable 55 is and whether I should invest my time and energy into it.

(You could also look at a copy of the math55a notes over vacation. I actually do not see much in the algebra that is covered in 55a that will be required for 55b, but Prof Auroux is the one to ask what exactly you should do to prepare.)

The only algebra necessary for 55B is basic group theory (definition of group, subgroup, homomorphisms, isomorphisms, etc.) which will be necessary for algebraic topology and which, luckily, I had the chance to learn in my first week of 55A.

I also think math 55, in these days, makes a serious pedagogical mistake, by cramming way too much material into much too short an amount of time to allow mastery.

Out of curiosity, what was 55 like back in your day? Did it cover less material?

good luck! you are in an enviable position, two good choices beckon.

Thank you so much for all of your advice!

 

[jedishrfu]

Which area of physics you're interested in from a short-term perspective might help you decide.

Would 55 be pertinent to Quantum Field theory or Cosmology or ...?

 

[Frabjous]

I've actually thought about this. In fact, I took 55A for a week and then dropped (I had zero proof-writing or abstract mathematics experience and had never taking a proof-based course before, so I was just unprepared for the rigors of 55; there is hope that 55B might work out for me after a semester of 25A). However, that one week of 55A was enough for me to see how much of a "superstar mathematician" Auroux is, to use your words. This is something that I missed throughout the rest of my semester in 25A. So yes, the two professors will definitely be a factor in my choice.

Do you feel that your mathematical maturity has significantly improved? I would plan on 25B, but try 55B to see if you can handle it. If you were only a math major, attempting 55B would be the obvious choice, but you also have physics to pay attention to.

 

[mathwonk]

@jedishrfu: Thank you. The book you mention was written in 1971, and I have no knowledge of it. It looks like a basic topology book, hence background for analysts, such as those professors were. Fairchild was Tulcea's student, according to "math genealogy"
https://www.mathgenealogy.org/id.php?id=4881

The basic such topology book in the 1960's was by John Kelley, General Topology.

Fairchild has no reported students, but Tulcea's students also include Robert Langlands of the famous and sweeping "Langlands' program", and his own math genealogy goes back to Weierstrass. Tulcea himself was student of the well known Einar Hille, author of a very impressive 2 volume work on complex analysis.

 

[mathwonk]

@Quantum55151:
The problem it seems was your complete lack of preparation for the sort of course math55 is, i.e. abstract and proofs based. That is hard to make up for in one semester. If you were advised into math 55; you must have a very strong record, and ability. Harvard seems to have a habit of assuming that being really bright and working really hard, can somehow overcome the natural amount of time and maturation that absorbing deep ideas often seem to require. Is auditing 55b a viable option while taking 25b for credit? That way you get graded by the (slightly) easier standard but get to enjoy the top level lectures.

There was no algebra in 55 the early days, it was a calculus course, primarily in Banach space and Hilbert space. The book by Loomis and Sternberg did not yet exist and the basic model was the famous book Foundations of Modern Analysis, by Dieudonne' (which I recommend).

Math 55 did Banach space differential calculus as in Dieudonne' or (as now in) Advanced Calculus by Loomis and Sternberg, and later covered finite dimensional integration (content theory, as in Loomis/Sternberg), the theory of bounded linear operators, spectral theory of compact hermitian operators in Hilbert space, Sturm-Liouville theory, and a brief introduction to manifolds and differential forms. So roughly chapters 1-8 of Loomis and Sternberg, (some of chapter XI of Dieudonne'), with maybe a lecture sketching the generalizations to manifolds in chapters 9-11. There was no complex analysis. The course covered somewhat less than others, since Loomis lectured more thoroughly than other professors.

I recommend Michael Spivak's little tome, Calculus on Manifolds, for finite dimensional advanced calculus, also Wendell Fleming's Calculus of several variables.

good luck to you. You have a bright future. Please don't neglect also to hear some of the great humanities professors at Harvard.

 

[jedishrfu]

Wow, thank you for the genealogy. I’ve found this to be true of my other professors and high school teachers as well. I feel I was amid greatness and didn't realize it.

Some of my high school music teachers were Big Band era musicians before they settled into teaching. It explained a lot of why our mediocre high school was able to win the state band championships year after year.

 

[Quantum55151]

Which area of physics you're interested in from a short-term perspective might help you decide.

I'm interested in high-energy theory. I don't know too much about it though, so I wouldn't be able to give you a specific area within HET.

 

[Quantum55151]

Do you feel that your mathematical maturity has significantly improved? I would plan on 25B, but try 55B to see if you can handle it.

I think my mathematical maturity has definitely improved. Has it improved to the extent that I will be able to succeed in 55B? That I will have to wait and see in January.

If you were only a math major, attempting 55B would be the obvious choice, but you also have physics to pay attention to.

Yes, that's also a factor because I am planning on taking a physics course that is not as hard as 55 but still nontrivial.

 

[mathwonk]

Pardon me for indulging in more suggestions.
Some suggested recommendations of books for math 55b topics.

For metric spaces, Dieudonne’ Foundations of modern analysis, including (a version of) Ascoli’s theorem. It also covers abstract differential calculus and inverse and implicit function theorems (in Banach spaces) and the Stone-Weierstrass theorem, if needed in that generality.

For the fundamental group and covering spaces, including Seifert Van Kampen (and classification of surfaces), I recommend Algebraic Topology, an introduction, by William Massey, (but he assumes you know what a Hausdorff topological space is).

For a source on the fundamental group that includes definition and properties of topological spaces, a good one is Andrew Wallace, An Introduction to algebraic topology, (but missing the all important Seifert- Van Kampen theorem for how to calculate them, and missing covering spaces, so not a substitute for Massey, but an augmentation).

I.e. read chapters 3 ,4 of Dieudonne' for metric spaces, then the first 3 chapters of Wallace for topological spaces, and then go to Massey for fundamental groups and covering spaces. This covers the first 17 lectures of math 55b.

For finite dimensional differential and integral calculus, Michael Spivak’s Calculus on Manifolds is excellent and brief. (It’s mostly not on manifolds until the very end, and omits Fourier series.).

Lang’s Analysis I, rewritten as Undergraduate Analysis, is also clear on many topics of differentiation and integration theory, e.g. characterizing the elementary real integral, inverse and implicit mapping theorems, and Stokes formula becomes clear forever once you see his simple explanation for a rectangle. He also treats Fourier series.

Stone-Weierstrass and Ascoli are in Dieudonne'. Lang's original book: Analysis I, from 1968, covered Stone-Weierstrass, but it was apparently omitted from the newer version: Undergraduate Analysis.

For complex analysis, my favorite is Henri Cartan’s Theory of analytic functions, including briefly Weierstrass P- (elliptic) functions and their Riemann surfaces. Ahlfors' book on complex analysis includes a chapter on metric and topological spaces, and is good on infinite products and elliptic functions, even if the late chapter on Riemann surfaces makes them seem harder and less intuitive than necessary, in my opinion.

The point is, if you start from a plane curve, of form F(z,w) = 0, then near any point (a,b) of this curve (where ∂F/∂w ≠0), w is locally an analytic function w(z) of z. The identity theorem for analytic functions implies that the whole curve is determined by just the one local power series expansion of w = w(z), in a disc on the z axis near z=a.
Ahlfors starts from a power series, constructs all the other power series obtainable from it, by "analytic continuation", and finally pastes them together to obtain a surface, which in the good case is the curve F=0. Doing it backwards like this makes it harder to imagine what is going on, and how natural it is, in my opinion. Excellent introductions to Riemann surfaces include the books by Rick Miranda, and by Phillip A. Griffiths.
https://www.amazon.com/Algebraic-Ri.../0821802682#customerReviews?tag=pfamazon01-20
https://www.abebooks.com/servlet/BookDetailsPL?bi=31903902729&searchurl=an=Phillip+a.+Griffiths&ds=30&rollup=on&sortby=20&tn=algebraic+curves&cm_sp=snippet-_-srp0-_-title2

and here is a link to notes for 55b when taught recently by Curt McMullen, that perhaps you already have; they seem to entirely omit the topology, other than metric spaces.
https://people.math.harvard.edu/~ctm/home/text/class/harvard/55b/10/html/home/course/course.pdf

cheeeezzz........., those are really condensed! Reminds me of the time I posted notes for a complete linear algebra course in 15 pages.
https://www.math.uga.edu/sites/default/files/inline-files/rev.lin_.alg_.pdf

or sketched the theory of Riemann surfaces in 5 pages:
https://www.math.uga.edu/sites/default/files/inline-files/Riemann.pdf

 

[mathwonk]

@Quantum55151:
I notice that very few undergraduate math courses at Harvard are taught by Professors. One of them however might be of interest to you (to audit) in the spring, Math 113, complex analysis, offered by the outstanding researcher and teacher Higgins Professor Joe Harris. This would help prepare you for the second half of the math 55b syllabus and make it easier to keep up on and learn that material. It might also touch on some of the topology concepts of the first part of 55b, such as the fundamental group, or at least homotopy of paths. The link is that, for complex analytic integrands, computing the path integrals over two paths that are "homotopic" to (deformable into) each other gives the same answer. In practice this allows one to choose a nicer path over which one can more easily compute an integral. One can prove the fundamental theorem of algebra this way, since a large loop parametrized by a polynomial of degree n, is homotopic to the one parametrized by z^n, which we know has n roots (suitably counted).

 

[Quantum55151]

Pardon me for indulging in more suggestions.
Some suggested recommendations of books for math 55b topics.

Thank you so much @mathwonk for all these resources!

One of them however might be of interest to you (to audit) in the spring, Math 113, complex analysis, offered by the outstanding researcher and teacher Higgins Professor Joe Harris.

Actually, an option that I was considering would be to simultaneously take 25b and 113 (both for credit) to get a 55-esque experience but with the material being taught at a slower pace. This has its pros and cons. 25b + 113 would probably be more conducive to learning; however, it would be inefficient in the sense of two psets every week instead of a single one in 55b and would also lack the topology component of 55b.

 

[mathwonk]

That actually might be a good plan. Joe is really outstanding, certainly a superstar. And the only topology in 55b is topology that is oriented towards complex analysis, so he might cover some of it. Homotopy of paths, for example, is defined in section II.1.6 in Cartan's analytic functions book, and covering spaces are essentially just mappings which are unramified in the sense of Cartan section VI.4.6.

E.g. the complex mapping z-->z^n, is unramified except at z=0, and (since also finite to one) is thus a degree n covering space of C-{0} ---> C-{0}.

The link with homotopy is that every path in the base of a covering space lifts to a path in the upper space, and homotopic paths in the base, lift to homotopic paths.

You might actually learn more this way, (in 25b + 113), since my instinct is that 55b is now designed as sort of a survey course, just sampling many subjects quickly.

To return to your earlier question, it looks almost to me as if, in my day, 55 was very abstract, and covered graduate level material, and that now it is maybe less advanced and combines but only sketches the content of several upper level undergraduate courses. So before it was maybe too advanced and abstract, and now it is perhaps too crammed and fast paced. The content however will probably be whatever the professor decides to cover.
(In those days there was a prior course, math 11, which covered rigorous one variable calculus, with theory, as in. Spivak's Calculus book, entirely proof based. The difference was that bright students then often had taken no calculus, so this was for the bright, un -calculused students.). So in the old days, math 11 was basically Spivak Calculus, plus whatever else the prof wanted to do in math 11b, and math 55 was a big chunk of the Loomis/Sternberg book. available free on Prof. Shlomo Sternberg's site:
'https://people.math.harvard.edu/~shlomo/docs/Advanced_Calculus.pdf

The main thing I learned in 55a is expressed in the first paragraph of the introduction to chapter 8 of Dieudonne'. Namely, the derivative of a function at a point, is a linear map approximating ("tangent to") that function near the point. Hence the derivative of a composition of functions is the composition of their linear approximations.

I also was proud of learning the definition of a derivative on a Banach space, but did not realize how hard it is to calculate them. In finite dimensions, they are calculated by partial derivatives, as Spivak makes clear.

By the way, one of my students was a phi beta kappa physics major at Harvard some decades ago, and took the math 25 sequence (from the famous superstar topologist Raoul Bott), so math 25 may serve you also quite well.

You might well discuss your choices with Professor Harris, who is very helpful and friendly.

 

[Quantum55151]

That actually might be a good plan. Joe is really outstanding, certainly a superstar. And the only topology in 55b is topology that is oriented towards complex analysis, so he might cover some of it.

Joe told me that with regard to topology 113 will be entirely self-contained -- all the necessary topology will be covered in the course.

You might actually learn more this way, (in 25b + 113), since my instinct is that 55b is now designed as sort of a survey course, just sampling many subjects quickly.

Yes, that certainly seems to be the case according to the reviews of past students -- a lot of material covered very fast and quite superficially. Although Auroux would beg to differ...

The main thing I learned in 55a is expressed in the first paragraph of the introduction to chapter 8 of Dieudonne'. Namely, the derivative of a function at a point, is a linear map approximating ("tangent to") that function near the point. Hence the derivative of a composition of functions is the composition of their linear approximations.

Could one see this as an intuitive explanation for the chain rule?

By the way, one of my students was a phi beta kappa physics major at Harvard some decades ago, and took the math 25 sequence (from the famous superstar topologist Raoul Bott), so math 25 may serve you also quite well.

Out of curiosity, do you know what math 25 was like back in your days?

You might well discuss your choices with Professor Harris, who is very helpful and friendly.

I have already talked to him! He said that all the options are very good. In his opinion, 55b has a slight edge over the 25b + 113 combo in that it covers algebraic topology in addition to real and complex analysis. At any rate, he said that I should shop all the courses and decide which ones I feel most comfortable in, which is probably what I'll do.

 

[mathwonk]

Sounds as if you have covered all the bases you can, and are well prepared to choose.

To your questions: yes that remark of Dieudonne' is exactly an explanation of why the chain rule is a completely natural result, properly formulated as a statement about maps. The Inverse function theorem also has a similar statement: If f is a smooth function from R^n to R^n, taking say 0 to 0, and if the derivative at 0, is invertible as a linear map, then f itself is invertible as a smooth map, on some neighborhood of 0. The implicit function theorem can also be stated: if the derivative f'(0) becomes a projection after a linear change of coordinates, then f also becomes a projection on some neighborhood of 0, after a smooth change of coordinates. (I strongly advise however also learning the usual old fashioned statement and applications of this theorem, for use in practice. In my case at least, after learning this abstract statement, I had no idea what Mumford meant when he said the implicit function theorem means, given an equation, you can use it to solve for some of the variables in terms of the other variables. )

In my day, there was no math 25. There was the 2 year 11/55 sequence, and the 3 year 1/20/105 sequence, for calculus, plus honors versions of 1/20/105. Math 11 "used" (i.e. recommended and totally ignored) Courant, later Spivak Calculus, Math 20 used a book like Angus Taylor, and 105 (or honors 105) used maybe Courant, or David Widder, while 55 was based on Dieudonne', or Apostol's Mathematical Analysis, later Loomis/Sternberg.
Apparently 25 was created as a middle ground between the two sequences, more sophisticated than 20/105, but less drastic than 55. There was also the changing situation in which virtually all strong high school students began to arrive with some calculus, whereas earlier very few had it available in high school.

AP calculus in high school caused a problem for teaching college math, since it creates a misconception in the students that they should be entitled to "advanced placement". But many high school "AP" students are not even prepared to skip one non honors calc course in college. If they try, one has these poorly prepared students in second semester calc, but not knowing the first semester well. This causes many good students to actually fail out of second semester or second year calculus. One way to adapt is by lowering the level of the college courses to match the weaker level of preparation.

"Advanced placement" also orients these students poorly, since they go from being honors high school students to taking advanced but non honors college classes. Thus their peer group is wrong.

At some schools, like UGa and Chicago, the old math 11 style "Spivak" course was kept for entering honors students, followed by an advanced version, based (at UGa) on Ted Shifrin's book:
https://www.amazon.com/Multivariabl...lus-Manifolds/dp/047152638X?tag=pfamazon01-20

Other schools like Stanford and Harvard, dropped the beginning Spivak course and plunged bright honors students right into a 55 style course, often with disastrous results. Rethinking these offerings perhaps resulted in the creation of math 25.
The physics major mentioned earlier, who started Harvard in math 25, had already prepared in high school with AP calculus followed by a proof based course from Marsden and Tromba's Vector Calculus book, augmented by an introduction to differential forms, (and taught by a college professor). He remarked that he "could not have survived" Bott's math 25 without the preparation in differential forms. (Check out the beautiful (advanced) book by Bott-Tu: Differential Forms in Algebraic Topology).

(If you choose or evaluate your courses based on the level of the professor, you begin to realize what it means to take beginning calculus as an AP course in high school, sometimes from someone who was not even a math major, but had only taken calculus in college.)

We have talked only about sources for advanced calculus, but I also recommend getting hold of a copy of Michael Spivak's Calculus, as a resource for one variable calculus, done right; or maybe better Courant, or Apostol's Calculus, since those books mention physics, unlike Spivak.
Here are used copies of both volumes of Courant, for under $20 total, a bargain.
https://www.abebooks.com/servlet/SearchResults?an=Courant&cm_sp=SearchF-_-home-_-Results&ref_=search_f_hp&sts=t&tn=differential and integral

Oh yes, and this discussion reminds of the third aspect of a college class: 1) professor, 2) syllabus, and 3) classmates! In terms of stimulating classmates, math 55b should probably be favored.
But looking again at the syllabi, that 25b syllabus is already quite extensive, i.e. rigorous one and several variable calculus, and the profs are both stars. So, and obviously the choice is up to you not me, but 25b + 113 looks to me like a more reasonable learning experience, at least on paper. So I guess the only way to decide is to try them yourself and get a feel, as you said. Good luck!

By the way, the stuff on axioms for real numbers in the 25b syllabus is covered in Dieudonne' chapters 1, 2. (And it seems pretty ambitious to cover all of the foundations of real numbers in just one 25b lecture, unless it is review.) If you can read that and maybe also chapter 3, you can likely handle anything theoretical you will see in these courses. I.e. Dieudonne' is the hardest source to read of all I mentioned. Here is a used copy for around $32. This is an amazing book for content, but very demanding reading (no pictures!).
https://www.abebooks.com/servlet/Se...Results&ref_=search_f_hp&sts=t&tn=foundations

 

[Quantum55151]

Sounds as if you have covered all the bases you can, and are well prepared to choose.
In my day, there was no math 25. There was the 2 year 11/55 sequence, and the 3 year 1/20/105 sequence, for calculus, plus honors versions of 1/20/105.

What I find interesting is that back in the day one would take 55 after taking a more elementary course like 11. Nowadays, 55 is only open to freshmen and so you are supposed to take it right out of high school which for many, including myself, is not an easy thing to do.

He remarked that he "could not have survived" Bott's math 25 without the preparation in differential forms. (Check out the beautiful (advanced) book by Bott-Tu: Differential Forms in Algebraic Topology).

Oh wow, 25 must have been watered down then, because nowadays differential forms are only taught in 55b. And even then, most of the class (from what I have been told) does not really understand them, because at the level of 55b, one lacks the mathematical tools to fully grasp the concept -- this kind of machinery is taught in a class like 132 (differential topology).

We have talked only about sources for advanced calculus, but I also recommend getting hold of a copy of Michael Spivak's Calculus, as a resource for one variable calculus, done right; or maybe better Courant, or Apostol's Calculus, since those books mention physics, unlike Spivak.
Here are used copies of both volumes of Courant, for under $20 total, a bargain.
https://www.abebooks.com/servlet/SearchResults?an=Courant&cm_sp=SearchF-_-home-_-Results&ref_=search_f_hp&sts=t&tn=differential and integral

Speaking of calculus/analysis books, where does Rudin's Principles of Mathematical Analysis fit into your classification? Is it of a similar calibre to Spivak and Apostol? You don't seem to mention it. Is there a reason why you prefer those other books to Rudin? I'm asking because 55b lists Rudin as the main (and only) source on real analysis.

Oh yes, and this discussion reminds of the third aspect of a college class: 1) professor, 2) syllabus, and 3) classmates! In terms of stimulating classmates, math 55b should probably be favored.

Yes, I can confirm that 55 has a better math community than 25. My first week in 55a was sufficient for me to realize that. On the other hand, in 55 one is surrounded by mathematical geniuses -- for example, this year there is one guy taking 55, 114 (integration and measure theory) and a graduate course on complex analysis -- and as a result it's quite easy to feel intimidated.

But looking again at the syllabi, that 25b syllabus is already quite extensive, i.e. rigorous one and several variable calculus, and the profs are both stars.

Yes, the 25b syllabus is definitely very solid.

So, and obviously the choice is up to you not me, but 25b + 113 looks to me like a more reasonable learning experience, at least on paper. So I guess the only way to decide is to try them yourself and get a feel, as you said. Good luck!

I think I would probably learn more from 25b + 113. That said, there is also the hypothetical option of doubling up on physics and taking two physics courses instead of one, in which case I would most likely have to limit myself to a single math course. I guess I need to evaluate my priorities and decide which of the two subjects -- math or physics -- I want to zoom in on this spring.

By the way, the stuff on axioms for real numbers in the 25b syllabus is covered in Dieudonne' chapters 1, 2. (And it seems pretty ambitious to cover all of the foundations of real numbers in just one 25b lecture, unless it is review.) If you can read that and maybe also chapter 3, you can likely handle anything theoretical you will see in these courses. I.e. Dieudonne' is the hardest source to read of all I mentioned. Here is a used copy for around $32. This is an amazing book for content, but very demanding reading (no pictures!).
https://www.abebooks.com/servlet/Se...Results&ref_=search_f_hp&sts=t&tn=foundations

Thank you so, so much @mathwonk for all this advice and all the links! I'll definitely check them out over winter break.

 

[mathwonk]

In 1960, math 11a covered roughly what is now on the syllabus for the first half of math 25b. It included axioms for the reals (handed out on day one), convergence of sequences and series, (real and complex) power series (applied to define exponential and trig functions), existence of maxima and minima of continuous real valued functions on closed bounded intervals, intermediate and mean value theorems, FTC, and probably more. I don't remember much from math 11b , but I do seem to recall coverage of vector spaces, including Hilbert and pre-Hilbert spaces, possibly Fourier series, and applications to differential equations.

Math 55 today is indeed very challenging for freshmen, but there some freshmen who have extremely strong preparation. Some years back when math55 was offered by Wilfried Schmid, the note taker had taken all the honors level college calculus courses at UGa, including the Spivak Calculus style course and the Shifrin multivariable calculus course, plus more advanced courses, while still in high school, and then took math 55 at Harvard, very well prepared. As I said, my physics student had taken one and several variable calculus in high school, including vector calculus from Marsden and Tromba, and had learned to compute with differential forms, and then even he took 25 rather than 55. It is not entirely clear to me what preparation is expected, or usual, for success in 55.

As to learning differential forms, there was a thread here on PF devoted to that topic some years back, and they went through a very nice book by David Bachman, available from him free online:
https://faculty.washington.edu/seattle/physics544/2011-lectures/bachman.pdf

As for Rudin's Principles of analysis book, it is famous for being very precise but very unmotivated, so I never recommend it for learning. But analysis professors love to recommend it, so you should take a look, maybe it will work for you. I prefer books by Spivak, Apostol, Berberian, Fleming, Lang, Simmons, and although quite difficult I admire Dieudonne'.


Here are some rather nice, and very down to earth, notes on differential forms, and their use in calculus, by Donu Arapura, that make them seem quite reasonable:
https://www.math.purdue.edu/~arapura/preprints/diffforms.pdf

and here is a very nice and accessible book on the topic, aimed at physical scientists. (I got over my own fear of differential forms in a little article by Flanders where he just taught me to calculate with them. being able to manipulate them made them seem less scary. Actually all they are is [combinations of and families of] determinants.)
https://store.doverpublications.com...Yftbzdo08qK2Zz3RT_1EAaXU51xIG5L5q1h1iHvx9Ordi

i.e. they are (linear combinations of) families of operators which, at each point, look at an ordered finite set of vectors and spit out the oriented volume of the parallelepiped those vectors (or certain of their components) span.

But don't be intimidated by wondering what they "are", just learn how they behave, i.e. learn to add, multiply, differentiate, and pull them back. When they give us axioms for real numbers these also only tell us how numbers behave, not what they are.

On the topic of the student body in math 55: yes one will naturally feel intimidated at times, maybe most of the time. I definitely felt that way when there. But the good part is that, if we can ignore that and not worry about how we compare to these geniuses, it turns out that we ourselves get stronger by being around them, as long as we do actually interact with them and open up to what they can offer us. I.e. when I later went elsewhere, it seemed my own level had improved from being around these exceptional people. In yoga, they speak of the aspect of someone who, although not a saint, has "been with saints". And they may be saying the same about you: "Here is this student who is taking these high powered math major courses and simultaneously majoring also in physics!"

If you want to test how much linear algebra you have learned in 25a, here is a revised set of the notes on introductory linear algebra I originally wrote for fun one Christmas break, first in 15 pages to see how short I could make it, and then expanded to 125 pages, as here. Determinants are only summarized. (The 15 page version is also on my website, linked above in post #27, if you are curious.)
https://www.math.uga.edu/sites/default/files/laprimexp.pdf

and here is another shorter (69pages) set of linear algebra notes from a course, with a complete treatment of determinants at the end.
https://www.math.uga.edu/sites/default/files/inline-files/4050sum08.pdf

 

[Muu9]

He says that I could definitely give 55B a try, but in his view, 55B is too superficial in its treatment of the material, especially of real analysis.

But isn't that okay with you, since you intend to be a physicist and not a mathematician?